associated prime

Let $R$ be a ring,and let $M$ be an $R$ -module.A prime ideal $P$ of $R$ is an for $M$ if $P={\\rm ann}(X)$ , the annihilator of some nonzero submodule $X$ of $M$ .

Note that if this is the case, then the module ${\\rm ann}_{M}(P)$ contains $X$ , has $P$ as its annihilator,and is a faithful (http://planetmath.org/FaithfulModule) $(R/P)$ -module.

If, in addition, $P$ is equal to the annihilator of a submodule of $M$ that is a fully faithful (http://planetmath.org/FaithfulModule) $(R/P)$ -module, then we call $P$ an of $M$ .

单词	AssociatedPrime
释义	associated prime Let $R$ be a ring,and let $M$ be an $R$ -module.A prime ideal $P$ of $R$ is an for $M$ if $P={\\rm ann}(X)$ , the annihilator of some nonzero submodule $X$ of $M$ . Note that if this is the case, then the module ${\\rm ann}_{M}(P)$ contains $X$ , has $P$ as its annihilator,and is a faithful (http://planetmath.org/FaithfulModule) $(R/P)$ -module. If, in addition, $P$ is equal to the annihilator of a submodule of $M$ that is a fully faithful (http://planetmath.org/FaithfulModule) $(R/P)$ -module, then we call $P$ an of $M$ .
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